# On the product dimension of clique factors

## TL;DR

This paper investigates the product dimension of disjoint unions of cliques, showing it grows logarithmically with the number of cliques for fixed clique size, and discusses open problems for simultaneous growth of parameters.

## Contribution

It extends previous results by establishing the asymptotic behavior of the product dimension for fixed clique size as the number of cliques grows, using linear algebra and Sperner capacity methods.

## Key findings

- Product dimension is approximately (1+o(1)) log2 r for fixed s.
- The asymptotic behavior when s and r grow together remains open.
- Combines linear algebra with Sperner capacity techniques.

## Abstract

The product dimension of a graph $G$ is the minimum possible number of proper vertex colorings of $G$ so that for every pair $u,v$ of non-adjacent vertices there is at least one coloring in which $u$ and $v$ have the same color. What is the product dimension $Q(s,r)$ of the vertex disjoint union of $r$ cliques, each of size $s$? Lov\'asz, Ne\v{s}et\v{r}il and Pultr proved in 1980 that for $s=2$ it is $(1+o(1)) \log_2 r$ and raised the problem of estimating this function for larger values of $s$. We show that for every fixed $s$, the answer is still $(1+o(1)) \log_2 r$ where the $o(1)$ term tends to $0$ as $r$ tends to infinity, but the problem of determining the asymptotic behavior of $Q(s,r)$ when $s$ and $r$ grow together remains open. The proof combines linear algebraic tools with the method of Gargano, K\"orner, and Vaccaro on Sperner capacities of directed graphs.